Volume 20, Number 5

Volume 20, Number 5, 2015 Nonholonomic mechanics

We derive the reduced equations of motion for an articulated $n$-trailer vehicle that moves under its own inertia on the plane. We show that the energy level surfaces in the reduced space are $(n + 1)$-tori and we classify the equilibria within them, determining their stability. A thorough description of the dynamics is given in the case $n = 1$.

Experimental Investigation of the Motion of a Body with an Axisymmetric Base Sliding on a Rough Plane

Abstract

In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.

We introduce a class of dynamical systems having an invariant measure, the modifications of well-known systems on Lie groups: LR and L$+$R systems. As an example, we study a modified Veselova nonholonomic rigid body problem, considered as a dynamical system on the product of the Lie algebra $so(n)$ with the Stiefel variety $V_{n,r}$, as well as the associated $\epsilon$L$+$R system on $so(n)\times V_{n,r}$. In the 3-dimensional case, these systems model the nonholonomic problems of motion of a ball and a rubber ball over a fixed sphere.

This paper is a review of the problem of the constructive reduction of nonholonomic systems with symmetries. The connection of reduction with the presence of the simplest tensor invariants (first integrals and symmetry fields) is shown. All theoretical constructions are illustrated by examples encountered in applications. In addition, the paper contains a short historical and critical sketch covering the contribution of various researchers to this problem.

Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors

Abstract

In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems. We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.